💡Alright, so I have this problem here where I'm given a point P(1, -2) on the terminal side of angle α, and I need to find the value of this trigonometric expression:frac{cosleft( frac{pi}{2} + alpha right) sinleft( -pi - alpha right)}{cosleft( frac{11pi}{2} - alpha right) sinleft( frac{9pi}{2} + alpha right)}Hmm, okay. Let me break this down step by step. First, since P(1, -2) is on the terminal side of angle α, I can use this point to find the sine and cosine of α. To do that, I remember that for any point (x, y) on the terminal side of an angle in standard position, the sine of the angle is y over r, and the cosine is x over r, where r is the distance from the origin to the point.So, let me calculate r. The distance formula is:r = sqrt{x^2 + y^2}Plugging in the values from point P:r = sqrt{1^2 + (-2)^2} = sqrt{1 + 4} = sqrt{5}Alright, so r is √5. Now, let's find sin α and cos α.sin alpha = frac{y}{r} = frac{-2}{sqrt{5}} = -frac{2sqrt{5}}{5}cos alpha = frac{x}{r} = frac{1}{sqrt{5}} = frac{sqrt{5}}{5}Got that. So, sin α is -2√5/5 and cos α is √5/5. Also, tan α would be y/x, which is -2/1 = -2. I might need that later.Now, looking at the expression I need to evaluate:frac{cosleft( frac{pi}{2} + alpha right) sinleft( -pi - alpha right)}{cosleft( frac{11pi}{2} - alpha right) sinleft( frac{9pi}{2} + alpha right)}This looks complicated, but maybe I can simplify each trigonometric function using angle addition formulas or periodicity properties.Let me tackle each part one by one.Starting with the numerator:1. **cos(π/2 + α):** I remember that cos(π/2 + α) is equal to -sin α. Let me verify that. Using the cosine addition formula:cos(A + B) = cos A cos B - sin A sin BSo, cos(π/2 + α) = cos(π/2)cos α - sin(π/2)sin α. We know that cos(π/2) is 0 and sin(π/2) is 1. So this simplifies to:0 * cos α - 1 * sin α = -sin αYep, that's correct. So, cos(π/2 + α) = -sin α.2. **sin(-π - α):** Hmm, sine is an odd function, so sin(-θ) = -sin θ. So, sin(-π - α) = -sin(π + α). Now, sin(π + α) is equal to -sin α because sine has a period of 2π and sin(π + α) = -sin α. So putting it together:sin(-π - α) = -sin(π + α) = -(-sin α) = sin αWait, let me double-check that. Starting with sin(-π - α):sin(-π - α) = sin(-(π + α)) = -sin(π + α) because sine is odd.Then, sin(π + α) is equal to -sin α because it's in the third quadrant where sine is negative. So:-sin(π + α) = -(-sin α) = sin αYes, that's correct. So, sin(-π - α) = sin α.So, the numerator simplifies to:cos(π/2 + α) * sin(-π - α) = (-sin α) * (sin α) = -sin² αAlright, moving on to the denominator:3. **cos(11π/2 - α):** This seems a bit tricky because 11π/2 is more than 2π. Let me see if I can subtract multiples of 2π to find an equivalent angle between 0 and 2π.11π/2 divided by 2π is (11/2)/2 = 11/4, which is 2.75. So, subtracting 2π (which is 4π/2) twice from 11π/2:11π/2 - 4π = 11π/2 - 8π/2 = 3π/2So, cos(11π/2 - α) is the same as cos(3π/2 - α). Now, let's use the cosine of a difference identity:cos(A - B) = cos A cos B + sin A sin BSo, cos(3π/2 - α) = cos(3π/2)cos α + sin(3π/2)sin αWe know that cos(3π/2) is 0 and sin(3π/2) is -1. So:0 * cos α + (-1) * sin α = -sin αTherefore, cos(11π/2 - α) = -sin α4. **sin(9π/2 + α):** Again, 9π/2 is more than 2π. Let's subtract multiples of 2π to bring it within 0 to 2π.9π/2 divided by 2π is (9/2)/2 = 9/4 = 2.25. So, subtracting 2π (which is 4π/2) twice:9π/2 - 4π = 9π/2 - 8π/2 = π/2So, sin(9π/2 + α) is the same as sin(π/2 + α). Using the sine addition formula:sin(A + B) = sin A cos B + cos A sin BSo, sin(π/2 + α) = sin(π/2)cos α + cos(π/2)sin αWe know that sin(π/2) is 1 and cos(π/2) is 0. So:1 * cos α + 0 * sin α = cos αTherefore, sin(9π/2 + α) = cos αSo, the denominator simplifies to:cos(11π/2 - α) * sin(9π/2 + α) = (-sin α) * (cos α) = -sin α cos αNow, putting it all together, the entire expression becomes:Numerator: -sin² αDenominator: -sin α cos αSo, the expression is:frac{-sin^2 alpha}{- sin alpha cos alpha}Simplify the negatives: both numerator and denominator have a negative, so they cancel out:frac{sin^2 alpha}{sin alpha cos alpha}Now, we can cancel one sin α from numerator and denominator:frac{sin alpha}{cos alpha}Which is equal to tan α.Earlier, I found that tan α = -2. So, the value of the expression is -2.Wait, let me just double-check all the steps to make sure I didn't make a mistake.Starting with the numerator:cos(π/2 + α) = -sin αsin(-π - α) = sin α (as we saw earlier)So, numerator is (-sin α)(sin α) = -sin² αDenominator:cos(11π/2 - α) = -sin αsin(9π/2 + α) = cos αSo, denominator is (-sin α)(cos α) = -sin α cos αThus, the expression is (-sin² α)/(-sin α cos α) = (sin² α)/(sin α cos α) = sin α / cos α = tan α = -2Yes, that seems consistent. So, the final answer is -2.**Final Answer**The value of the given expression is boxed{-2}.

💡Okay, so I have this equation: z multiplied by i equals -1 plus three fourths i. I need to find the value of z. Hmm, complex numbers can be a bit tricky, but let's see if I can figure this out step by step.First, I know that z is a complex number, which means it can be written in the form a + bi, where a and b are real numbers. So, z = a + bi. The equation given is z * i = -1 + (3/4)i. I need to solve for z, which means I need to isolate z on one side of the equation.To do that, I can divide both sides of the equation by i. That should give me z by itself. So, z = (-1 + (3/4)i) / i. Now, dividing by i isn't something I do every day, so I need to remember how to handle that. I recall that dividing by i is the same as multiplying by -i because i * (-i) = 1. So, maybe I can multiply both the numerator and the denominator by -i to simplify the expression.Let me try that. If I multiply the numerator (-1 + (3/4)i) by -i, I get (-1)*(-i) + (3/4)i*(-i). Let's calculate each part separately. First, (-1)*(-i) is just i. Then, (3/4)i*(-i) is (3/4)*(-i^2). I remember that i squared is -1, so this becomes (3/4)*(-(-1)) which is (3/4)*(1) = 3/4. So, putting it all together, the numerator becomes i + 3/4.Now, the denominator is i*(-i). Again, i*(-i) is -i^2, which is -(-1) = 1. So, the denominator simplifies to 1.Therefore, z = (3/4 + i) / 1, which is just 3/4 + i. Wait a minute, that doesn't seem right because when I started, the equation was z * i = -1 + (3/4)i, and if z is 3/4 + i, then multiplying by i should give me -1 + (3/4)i. Let me check that.Multiplying z = 3/4 + i by i: (3/4 + i)*i = 3/4*i + i*i = (3/4)i + i^2. Since i^2 is -1, this becomes (3/4)i - 1, which is -1 + (3/4)i. Okay, that matches the original equation, so it seems correct.But wait, in my initial calculation, I ended up with z = 3/4 - i, but when I checked, it turned out to be 3/4 + i. Did I make a mistake somewhere? Let me go back through my steps.Starting again: z * i = -1 + (3/4)i. To solve for z, divide both sides by i: z = (-1 + (3/4)i)/i. To simplify, multiply numerator and denominator by -i: [(-1 + (3/4)i)*(-i)] / [i*(-i)].Calculating the numerator: (-1)*(-i) = i, and (3/4)i*(-i) = (3/4)*(-i^2) = (3/4)*(1) = 3/4. So, numerator is i + 3/4.Denominator: i*(-i) = -i^2 = 1. So, z = (3/4 + i)/1 = 3/4 + i. But earlier, I thought I had 3/4 - i. Hmm, maybe I confused the signs somewhere.Wait, let's double-check the multiplication in the numerator. (-1 + (3/4)i)*(-i) = (-1)*(-i) + (3/4)i*(-i) = i + (3/4)*(-i^2). Since i^2 is -1, this becomes i + (3/4)*(1) = i + 3/4. So, it's definitely 3/4 + i.But when I checked by multiplying z = 3/4 + i by i, I got -1 + (3/4)i, which is correct. So, why did I initially think it was 3/4 - i? Maybe I made a mistake in the sign when multiplying.Wait, no, the calculation seems correct. So, perhaps the initial thought of 3/4 - i was wrong, and the correct answer is 3/4 + i. But let me try another approach to confirm.Another way to solve z * i = -1 + (3/4)i is to represent z as a + bi and then solve for a and b. So, let's set z = a + bi. Then, z * i = (a + bi)*i = ai + bi^2 = ai - b, since i^2 = -1.So, z * i = -b + ai. According to the equation, this equals -1 + (3/4)i. Therefore, we can equate the real and imaginary parts:Real part: -b = -1Imaginary part: a = 3/4From the real part, -b = -1 implies b = 1. From the imaginary part, a = 3/4. Therefore, z = a + bi = 3/4 + i.So, this confirms that z = 3/4 + i. Earlier, I thought I had 3/4 - i, but that must have been a mistake in the sign during the initial calculation. It's important to carefully check each step, especially when dealing with complex numbers and their properties.In summary, by either dividing both sides by i and simplifying or by expressing z as a + bi and equating parts, we arrive at the solution z = 3/4 + i. It's always good to verify the solution by plugging it back into the original equation to ensure it holds true.

💡First, I need to determine the number of ways to choose two socks of the same color from each color category.For the white socks, there are 5 socks, and I want to choose 2. This can be calculated using the combination formula:[binom{5}{2} = frac{5 times 4}{2 times 1} = 10]Similarly, for the brown socks, there are also 5 socks:[binom{5}{2} = frac{5 times 4}{2 times 1} = 10]For the blue socks, there are 3 socks:[binom{3}{2} = frac{3 times 2}{2 times 1} = 3]And for the red socks, there are 2 socks:[binom{2}{2} = frac{2 times 1}{2 times 1} = 1]Finally, I add up all the possible ways to choose pairs of socks of the same color:[10 (text{white}) + 10 (text{brown}) + 3 (text{blue}) + 1 (text{red}) = 24]

💡First, I need to determine the number of students in the class based on the given average ages.Let's denote the number of students as ( n ).The total age of all the students is ( 22n ) years.When the teacher's age of 46 years is included, the average age increases by 1 year to 23 years. The total number of people now becomes ( n + 1 ).The total age with the teacher included is ( 22n + 46 ) years.Using the new average age, I can set up the equation:[frac{22n + 46}{n + 1} = 23]Next, I'll solve for ( n ):[22n + 46 = 23(n + 1)]Expanding and simplifying:[22n + 46 = 23n + 23][46 - 23 = 23n - 22n][23 = n]Therefore, there are 23 students in the class.

💡First, I recognize that when 120 is divided by ( j^2 ), the remainder is 12. This implies that ( j^2 ) must be a divisor of ( 120 - 12 = 108 ).Next, I factorize 108 to find its divisors:[108 = 2^2 times 3^3]The square factors of 108 are 1, 4, 9, and 36. Among these, only 36 leaves a remainder of 12 when dividing 120:[120 div 36 = 3 text{ with a remainder of } 12]Thus, ( j^2 = 36 ), which means ( j = 6 ).Finally, to find the remainder when 180 is divided by ( j = 6 ):[180 div 6 = 30 text{ with a remainder of } 0]

💡Okay, so I have this problem here: If the graph of the function ( y = f(x) ) passes through the point ( (1, 3) ), then the graph of the function ( y = f(-x) + 1 ) must pass through which point? Hmm, let me try to figure this out step by step.First, I know that when dealing with function transformations, certain changes to the function affect the graph in specific ways. For example, replacing ( x ) with ( -x ) reflects the graph over the y-axis, and adding a constant outside the function shifts the graph vertically.So, starting with the original function ( y = f(x) ), which passes through ( (1, 3) ). That means when ( x = 1 ), ( y = 3 ). In other words, ( f(1) = 3 ).Now, let's consider the transformed function ( y = f(-x) + 1 ). I need to find a point ( (a, b) ) that lies on this new graph. To do this, I can think about how the transformations affect the original point ( (1, 3) ).First, the reflection over the y-axis. Reflecting a point over the y-axis changes the x-coordinate's sign. So, reflecting ( (1, 3) ) over the y-axis would give me ( (-1, 3) ). That makes sense because reflecting over the y-axis doesn't change the y-value, just flips the x-value to the opposite side.Next, there's a vertical shift of 1 unit upward. Adding 1 to the entire function ( f(-x) ) shifts every point on the graph up by 1. So, taking the reflected point ( (-1, 3) ) and shifting it up by 1 would result in ( (-1, 4) ).Wait, let me double-check that. If I plug ( x = -1 ) into ( f(-x) ), that would be ( f(-(-1)) = f(1) ), which we know is 3. Then, adding 1 gives ( y = 3 + 1 = 4 ). So, yes, when ( x = -1 ), ( y = 4 ). That means the point ( (-1, 4) ) is on the graph of ( y = f(-x) + 1 ).Is there another way to think about this? Maybe by considering the transformations in reverse. If I have ( y = f(-x) + 1 ), to find a corresponding point, I can set up the equation ( b = f(-a) + 1 ). Since I know ( f(1) = 3 ), I can set ( -a = 1 ) to use this known value. Solving for ( a ) gives ( a = -1 ). Then, plugging back into the equation, ( b = 3 + 1 = 4 ). So, again, the point is ( (-1, 4) ).I think that confirms it. The transformations applied to the original function affect the coordinates in a predictable way. Reflecting over the y-axis changes the sign of the x-coordinate, and shifting up affects the y-coordinate by adding the shift value.Just to make sure I'm not missing anything, let me visualize it. If I have the original point ( (1, 3) ), reflecting it over the y-axis would move it to ( (-1, 3) ). Then, moving it up by 1 unit would bring it to ( (-1, 4) ). Yep, that seems right.Another thing I can consider is plugging in different values to see if it makes sense. For example, if ( x = 0 ), then ( y = f(0) + 1 ). But since I don't know ( f(0) ), that doesn't help me directly. However, since I know ( f(1) = 3 ), I can use that specific point to find the corresponding point on the transformed graph.So, in summary, the transformations applied to ( f(x) ) to get ( f(-x) + 1 ) are a reflection over the y-axis and a vertical shift upward by 1 unit. Applying these transformations to the point ( (1, 3) ) gives me the point ( (-1, 4) ) on the new graph.I think I'm confident with this answer now. It makes sense both algebraically and graphically. The key was understanding how each transformation affects the coordinates of the points on the graph.**Final Answer**The graph of the function ( y = f(-x) + 1 ) must pass through the point with coordinates boxed{(-1, 4)}.